If the usual floor cleaners don't do the trick try some mineral spirits on a rag to wipe away the stains. It's amazing what that stuff cleans up on floors such as this.
If you do this fairly regularly you'd be wise to invest in a patch of tough hallway runner carpet. The stuff with the non skin rubber backing. It's fairly cheap so after some years of use when it's totally stained and grotty you can toss it out with a clear conscience and get another one. Frankly newspapers protect so little and transfer so much ink onto everything they touch that it's a waste of time using them.